| Budget Amount *help |
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Research Abstract |
The dynamical systems theory originated from two notions of hyperbolicity and structural stability, and researches toward a solution to the stability conjecture played an important role in the developments of the theory. The conjecture asserts every structurally stable system is hyperbolic, and in 1987 it was proved by Mane for r=1. In the proof, so-called Franks lemma was essential. Since the lemma does not work for the C^r-topology (r【greater than or equal】2), the conjecture is still open when r【greater than or equal】2. The purpose of this research project is to prove the hyperbolicity under the shadowing-C^r-open condition (r【greater than or equal】2) fusing with Pesin theory, and by applying the techniques obtained in this process, we try to solve the C^r-stability conjecture for r 【greater than or equal】 2.
In 2003, we restrict ourselves to 2-dimensional dynamical systems and concentrated to prove the hyperbolicity of the system under the shadowing-C^r-open condition. In 2004, we continuously proceeded the above strategy, but there were noting special for publication. However, for some partial results obtained in this research, we have found some handles to generalize them for higher dimensions. In our opinion, the achieve percentage of this project might be evaluated 50%.
Before to show the hyperbolicity of the dynamical systems, it is necessary to prove the hyperbolicity of the periodic points. Under the shadowing-C^r-open condition (r【greater than or equal】2), the head investigator proved the hyperbolicity of the periodic points etc., and making use of the facts, he also proved the hyperbolicity for 2-dimensional dynamical systems by assuming additional conditions (as was stated it turned out that this result can be generalized).
A base of this research project has been completed. Hereafter we would like to do my best to complete the project.
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